"Bach-style keyboard tuning"

Article by Mark Lindley and Ibo Ortgies, November 2006 issue of Early Music,
published on the web as
"advance access version October 3 2006".

This is an eleven-page pre-print version, as a PDF file for download from Oxford's web site.
(Subscription or library access are necessary to
download this file
(cal065v1); I
am not at liberty to re-distribute it directly....)

The chain of reasoning in Bradley Lehman's article in Earlymusic, xxxiii (2005), pp.3-23, 211-31, is full of weak links.The notion that Bach followed a mathematical rule when tuningis contrary to relevant documentary evidence that he did notgo in for theoretical treatments and that mathematizingwould never have led to success in ensuring the execution ofan unobjectionable temperament. An expert and musicianlytuner would indeed temper alike the 5ths C-G-D-A-E, but wouldnot feel obliged (as Lehman imagines) to temper some 5ths exactlytwice as much as others. Lehman's premise that a mathematicallyrigid tuning-scheme is hidden cryptically in a decorative scrollon the title-page of WTC I is daft, and his belief that thereis only one musically feasible way to interpret this allegedevidence is disproved by the existence of several other suchways besides his (which is based on misreading a serif as aletter). Lehman misrepresents Sorge's account of a certain theoreticalscheme from after Bach's death (which Lehman regards as evidenceapplicable to Bach). No tuning-theorist close to Bach approvedof tempering E-G# as much as Lehman does. Lehman's ideathat Bach's secret tuning is uniquely beautiful for music byFrescobaldi et al. is outlandish. That his wife agrees withhim is not substantial evidence as to 17th- or 18th-centurytaste. His one-size-fits-all approach obscures some relevantfacts about church-organ tuning in those days. If Bach advisedsome organ builders about tuning, Zacharias Hildebrandt wouldbe the most likely one, but the meaning of the statement byBach's son-in-law that Hildebrandt followed Neidhardt,while clearly ruling out Lehman's scheme, is unclear in someother ways since some of Neidhardt's ideas about tempered tuningchanged over the years.

My initial remarks, October 2006:

I will present a fuller response to this article after it appears in print with its final page numbers etc.

How do these authors presume to be inside my head, asserting what I allegedly "imagine" and "believe", and planting
onto me the "outlandish" idea that "Bach's secret tuning is uniquely beautiful for music by Frescobaldi et al" (which of course I've not said, and don't believe...)? On the way to labelling my work as "daft"?

Well, four additional pages of my comments are already drafted, to deal with such problems.
Their article in general tilts at a series of
straw-man windmills
against my article: imputing to me
a series of "implicit premises" that I didn't say and don't necessarily believe
(at least in the overstated way they present my alleged ideas!),
and then proposing to knock them down as "weak links". They recycle some of my reasoned
(and I hope, carefully demonstrated!) conclusions from the musical evidence as if those were premises, thereby mis-representing the developmental direction of my arguments....
We'll come back to all that.

Drs Lindley and Ortgies discard Bach's WTC title-page drawing as non-evidence, but only after they've asserted
it's a "daft" pursuit altogether; and they do this in a context of presenting several other
speculative temperament-interpretations of the same drawing.
Are those other ones equally daft, more daft, less daft than mine;
and how is such a level of alleged daftness to be judged objectively?
Their point, obviously, is that all such speculation isn't
reasonably reliable enough to them within their own historical and musical epistemology...but then they
violate their own game, going on in this article to present a "Bach-style"
speculative temperament of their own devising, with no documentary evidence, and no
supported arguments describing the process by which they've concocted it!
We'll come back to that, too.

The authors even try to dismiss Bach's extant music as non-evidence
(or at least as insufficiently reliable, as to performance practices), alleging that 18th century
musicians didn't actually expect to play it. (And how do they know what 18th century
people did not do, trying to "prove" that negative case?!...) They assert:
"No extant documents from Bach's lifetime say that he (or anyone else) performed on an organ
his written keyboard compositions; the reports of his playing refer apparently to improvisation."
Does this constitute some manner of proof that Bach and everybody else did not use any
of that extant music? And that we dare not, therefore, take any inferences whatsoever about
keyboard-tuning practices, drawn from the actual music that survives from that milieu?
Not even from play-throughs listening to Das wohltemperirte Clavier itself, the book that I put forth
as primary evidence both musical and graphical?

With all this evidence being kicked out the window, what are we to be left with?
Dump all bathwater and all babies!

All of which is to point out: my presentation, my writing style, and my consideration of evidence
all do not fit into their paradigm...and Drs Lindley and Ortgies are then trying to disprove
my work within that paradigm (rather than taking my work on its own terms, thinking outside their box). Incidentally, it was my observation of anomalies within that paradigm
that got me seriously interested in this project, in early 2004, serving as what I felt to be a
breakthrough: my sense that their paradigm
happened to be untenable, because of the musical evidence that contradicts some of its points!
The standard Lindley assertions, in print, about the sizes of major 3rds Db-F and Ab-C had already
alerted me, for many years preceding, that something was possibly wrong with the historical paradigm as
he and others have presented it. It sounded wrong to me, directly in the music; and I couldn't believe that
nothing better (as to a euphonious result) was ever available to Bach's genius...why would Bach write
ugly music with any horrible intervals in it?
So, with tuning lever in hand and playing the music, I began to think and work outside the
limitations of their methodologies: to see what else might be possible (and yet credible and
reasonable, backed up with what I believe to be sufficiently hard evidence).

For now, some of the most immediately important remarks of mine
are as follows, as the Lindley/Ortgies article contains a number of factual errors.
Several of those look merely like typesetting/proofreading glitches, and scarcely affect their arguments.
One of them, however, is a much profounder faux pas of omission on their part...and
it's a straightforward example of discarding (or at least overlooking) important evidence,
not tied to assessments of anyone's taste.

First, the several numerical errors in their charts. My suggested corrections here are to ensure both
the internal consistency within each chart (i.e. the sizes of 5ths are just plain wrong, according to the sizes
of the 3rds!), and consistency with their explanation in the body of their article.

All three of these charts contain typesetting errors (proofreading errors?) among the sizes of the 5ths.

In figure 2, equal temperament, the lower right numeral between Bb and F should obviously be 1, not -1.

In figure 5, the total of their twelve 5ths is 12.5, whereas (in all temperaments) it must be 12,
using their present units of measurement. The
error is at the same place here, the numeral 1/2 between Bb and F...which should be 0, according to their
3rd-calculations here. Their text does not explain the rationale for selecting a pure 5th here,
or any other size in particular, but that's beside the point.

In figure 6, their total of 5ths is 13 this time. Furthermore, their text explains that this diagram
represents a simple and entire transposition of my temperament. Well then, if so, there are two errors
among their 5ths. The numeral between A and E should be 2 rather than 1; and the numeral between F and C
should be -1 rather than 1. Their text also remarks that this particular temperament would create the
effect of ending both books of WTC on "the most heavily tempered" triad of the temperament, namely B major.
Interesting, and true, but so what?

Now, on to the most serious error of omission. In one of the sections of their article,
Lindley and Ortgies assert the following. The part in [brackets] here is their presumptive assertion
(i.e. straw-man argumentation)
against what they believe to be a premise of mine, and then it's followed by their assertive refutations:

"[Significant documentary support for the claim that Bach always tempered E-G# by as much as shown in fig. 3 is to be found in some remarks by Sorge ...] Neither Sorge nor Neidhardt ever countenanced tempering E-G# (for any reason whatever) by as much as Dr Lehman says Bach always tempered it. Neidhardt would never countenance tempering E-G# more than Ab-C."

"Ever?" (Whether in print or not...?) "For any reason whatever?" Strong statements. And, in their
abstract it corresponds with this sentence of theirs:
"No tuning-theorist close to Bach approved of tempering E-G# as much as Lehman does."

Here's a simple counter-example in which Neidhardt did both these things, in print: making
E-G# tempered as widely as mine, and making it wider than Ab-C.

Without my further analytical comments yet at this point, this is one of
Neidhardt's 21 example temperaments published in his 1732 treatise.
(Neidhardt, Johann Georg. Gäntzlich erschöpfte, mathematische Abtheilungen des
diatonisch-chromatischen, temperirten Canonis Monochordi, Königsberg and Leipzig 1732.)

This musically interesting temperament with three slightly wide 5ths is #11
among the twelve "Quinten-Circuls" presented by Neidhardt as a set of charts.
This is one of five(!) example temperaments in that Neidhardt treatise that each
"countenance tempering E-G# by as much as" the
E-G# in my presented hypothetical Bach temperament.
Also, this one happens to have E-G# noticeably wider than Ab-C!

Its 5ths are as follows,
using Neidhardt's standard (and theoretically indivisible--to him and Sorge but not to Lindley!)
units of 1/12 Pythagorean comma:

C 1 G 2 D 3 A 1 E -1 B 1 F# 2 C# -1 G# 3 D# 1 Bb -1 F 1 C.

This chart is generated by the same
spreadsheet I use for all of my larips.com analyses, for consistency.
Look especially at the C-E-G#/Ab-C column at the bottom, showing the sizes of stacked major 3rds measured in
1/11 fragments of the syntonic comma -- the same unit of analytical measurement
later used by Sorge,
correcting/updating
Neidhardt's own slightly incorrect 1/12 method of reckoning from this same 1732 document (and Neidhardt's
1724 treatise as well).

Here in Neidhardt's 5th-circle temperament #11:
C-E is the smallest at size 4, Ab-C is next at size 7
(the same size as in equal temperament), and E-G# is widest at size 10 (where 11 would be a full syntonic
comma sharp). In my temperament, for comparison, the C-E is size 3, the Ab-C is 8, and E-G# is this same 10.

I have no explanation why Lindley and Ortgies have overlooked/omitted the existence of this Neidhardt
temperament...or the four
others in that same 1732 Neidhardt document that also have an E-G# as wide as (or wider than!) mine.
Three of these five temperaments have been readily available in J Murray Barbour's 1951 book (Tuning and Temperament: A Historical Survey), at tables 172, 173, and 175; probably the most popular and widely-used
book in the English-language literature about tuning!
And, a facsimile scan of Neidhardt's chapters (along with a reliable ASCII transcription) has been available
since December 2005 at http://harpsichords.pbwiki.com/Tuning .
I myself discussed the features of this particular temperament #11 in public,
6 March 2006,
in the HPSCHD-L discussion group where Dr Ortgies is a regularly participating member....

See also Neidhardt's final temperament among this batch, the 3rd-Circle #5.
(It is available as the final page of this set of analytical tables presenting all 21 of Neidhardt's 1732 temperaments.)
It is a quite moderate temperament
all around, where all the major 3rds are of sizes 6, 7, or 8, and therefore it sounds like pseudo-equal
temperament. And yet: its E-G# is larger (size 8) than its Ab-C (size 7). This temperament, as well,
goes directly against the Lindley/Ortgies assertion:
"Neidhardt would never countenance tempering E-G# more than Ab-C."

In total, six of the 21 Neidhardt examples published in 1732 break one or both of the Lindley/Ortgies points
about the positioning of the note G#/Ab, from E or C on either side of it.

Neidhardt saw fit to publish these several temperaments
as part of his pedagogical discussion of temperament design strategies (which is a point of his 1732
piece!). In an attempt to refute my work, or at least to cast doubt against it, it simply doesn't do
for Ortgies and Lindley to assert that this Neidhardt precedent of an E-G# size
(also being wider than Ab-C!) didn't exist;
the extant documentation is right here in
a facsimile of Neidhardt's chapters, and analyzed in greater detail
here.
Especially so, as their own assertions are that I've omitted anything that should be important
within their epistemology. They volunteered to play defense here against my work, so let's examine
their manner of defense, along with the factual (or not!) statements of evidence.
Shouldn't they play by the same rules they are proposing to hold me to?

Lindley and Ortgies in their article offer no reaction at all to my five
"web supplement" files that are integral
parts of the original Oxford article; as if only the on-paper sections of that
article are worthy of consideration?
I notice also that they haven't engaged any of the technical material from my printed part 2,
which has been in print for 16 months (May 2005 issue of Early Music, and downloadable for
free here). Their reactions are almost entirely to
the presentation in part 1 (February 2005),
which is only a small portion of my argument; and which they evidently don't fancy enough to
take the rest of the presented argument seriously.

They have not acknowledged or engaged any of my 2005 article
from Clavichord International, in its clarifications of the material;
have they read it? (It's more widely available on the web
now, to facilitate any such discussion!)
Nor have they offered any musical or historiographical
reaction to any of my (or other people's)
recordings that use my proposed temperament;
isn't this too a dodging of potentially interesting evidence, on their part?
(My recordings, too, are freely available as samples on the internet: serving as musical examples
that go along directly with my printed analyses.
[Harpsichord]
[Organ])

To put this bluntly: does my putative Bach temperament actually
do what I claim it does, in the music where I suggest it's most relevant, which is the
musical and historical hypothesis to be engaged? Without engaging these materials
or discussing Bach's music, it
seems to me that Drs Ortgies and Lindley are merely picking at my writing style,
and my refusal to restrict myself to the only methodological paradigm they will accept.

In their roster of "implicit premises" imputed to my work,
they have set up a straw-version of my part 1, in isolation: to be knocked down without
apparently taking the other sections of the presented argument and evidence into account.
I can see that their list of premises is a convenient framework on which their own
article is constructed, to organize their material; fine.

Still, rather than guessing at my "implicit premises", they could have engaged my
publicly stated premises and assumptions that have been
available for the record since 31 May 2006.
I presented those points directly in a HPSCHD-L group discussion,
and I clarified those ideas further through the first weeks of June
2006. But, such discussion has already been going vigorously for much longer than
that, since February 2005: with my cards face up on the table,
and my open invitations to play and discuss the music
(which is, after all, where it matters in playing Bach!).

This is all on public record,
in the hope that the musical and historical material in my hypothesis
might actually be engaged for discussion. Such topics include key characters,
the roles of 18th century pedagogy (keyboard and otherwise--dealing especially with 1/6 comma
regular and irregular systems!), the measurement of competing
temperament systems in Bach's milieu, and Bach compositions (whether inside or outside the WTC) that
have always presented special problems of intonation...or any special effects
aligning with their vocal texts.
But, judging from the range of remarks by Ortgies and Lindley here, it's almost
as if I never brought up any of these important issues as important parts of my argument!
The whole venture is dismissed as "daft" and therefore outside their consideration.

In this long-awaited article by Lindley and Ortgies, I hoped they would address these technical issues
of the topic, and take my hypothesis under serious consideration. Instead, though, it appears they've
merely offered some (wrong) guesses into my imagination (thereby reducing their own presentation to
straw-argumentation), and have dismissed my hypothesis outright without bothering to take
all of it into account!
In the 16 months since May 2005, they have had plenty of time--and plenty of additional
public information from me--to study my published hypothesis carefully, thoroughly,
and to construct a much more substantial response/dialogue than this article offers. Why haven't they?
Instead, we're treated to merely patronizing barbs such as this one by them (page 6 of this
article):

"A modern harpsichordist's historical misjudgments are no great moral lapse, however. The
music belongs to Dr Lehman just as much as to us and we wish him well in performing it."

Um, thanks.

As noted above: my fuller and more tediously heavy-handed comments about all this (and more)
will be updated to this space, after their article appears in print.
I have another four or five pages of my blathering-on about this, to come in the next month or two.

Meanwhile, I have notified the publisher and authors of the factual problems with their
tables and their Neidhardt assertions, in the hope that those can be corrected before the print!

My remarks on their final version, November 2006:

Their November abstract has changed slightly since October, becoming this:

The chain of reasoning in Bradley Lehman's article in Earlymusic, xxxiii (2005), pp.323, 21131, is full ofweak links. The notion that Bach followed a mathematical rulewhen tuning is contrary to relevant documentary evidence thathe did not go in for theoretical treatments andthat mathematizing would never have led to success inensuring the execution of an unobjectionable temperament.An expert and musicianly tuner would indeed temper alike the5ths CGDAE, but would not feel obliged(as Lehman imagines) to temper some 5ths exactly twice as muchas others. The premise that a mathematically rigid tuning-schemeis hidden cryptically in a decorative scroll on the title-pageof WTC I is daft, and Lehman's belief that there is only onemusically feasible way to interpret this alleged evidence isdisproved by the existence of several other such ways besideshis (which was based on misreading a serif as a letter). Lehmanmisrepresents Sorge's account of a certain theoretical schemefrom after Bach's death (which he regards as evidence applicableto Bach). No tuning-theorist close to Bach approved of temperingEG# as much as Lehman does. Lehman's idea that Bach'ssecret tuning is uniquely beautiful for music by Frescobaldiet al. is outlandish. His one-size-fits-all approach obscuressome relevant facts about church-organ tuning in those days.If Bach advised some organ builders about tuning, ZachariasHildebrandt would be the most likely one, but the meaning ofthe statement by Bach's son-in-law that Hildebrandt followedNeidhardt, while clearly ruling out Lehman's scheme,is unclear in some other ways since some of Neidhardt's ideasabout tempered tuning changed over the years.

See also
the page of errata
posted by Dr Ortgies onto
HPSCHD-L, November 10 2006,
to accompany the release of the article; it is also available
at his
own web site.
Indeed that page of errata acknowledges and corrects the technical points I noted above (my
October section here about the table typos, and the Neidhardt E-G# business),
...while not acknowledging this proofreading assistance back to me.... At least they have changed
their two unsupportable statements about Neidhardt, explaining their points now with better sentences there!

I note several changes in the abstract itself:

"Lehman's premise that a mathematically rigid tuning-scheme is hidden cryptically in a decorative scroll on the title-page of WTC I is daft, and his belief that there is only one musically feasible way to interpret this alleged evidence is disproved by the existence of several other such ways besides his (which is based on misreading a serif as a letter). Lehman misrepresents Sorge's account of a certain theoretical scheme from after Bach's death (which Lehman regards as evidence applicable to Bach)."

"The premise that a mathematically rigid tuning-scheme is hidden cryptically in a decorative scroll on the title-page of WTC I is daft, and Lehman's belief that there is only one musically feasible way to interpret this alleged evidence is disproved by the existence of several other such ways besides his (which was based on misreading a serif as a letter). Lehman misrepresents Sorge's account of a certain theoretical scheme from after Bach's death (which he regards as evidence applicable to Bach)."

At least this set of small revisions to the abstract acknowledges several things:

That the allegedly "daft" thing isn't "Lehman's premise". They're now painting with the broader brush here,
that anyone who seeks a temperament in this drawing (i.e. using that premise) is doing something "daft".

That the "misreading a serif as a letter" is not a make-or-break point of my hypothesis (and never was).
My "misreading" of an object looking like a letter C was merely a catalyst for me to experiment with
temperament layouts that have middle C in that particular position. But my primary reason is,
and remains,
my observation that the entire C major scale is tuned first on the keyboard,
as a set of regular naturals (except B), which necessitates beginning the line with F;
and C indeed falls into that position where that "serif" happens to look like a C. Coincidence?

This passage in their abstract is still rife with straw-man assertions,
since I don't in fact hold such a belief about
"only one musically feasible way to interpret this alleged evidence" (a second one by me has been
available as "Bonus 3" here among my LaripS.com pages, since March 29 2006, and posted
elsewhere on March 8 2006),
and since I don't believe that any of this must be "mathematically rigid".
Lindley and Ortgies are imputing to me things I didn't say.

Other parts of the abstract and article are similarly problematic, in the way they misrepresent my
presented material and my "ideas" and "beliefs", on the way to taking shots at those points.

Let me be clear about this: I do not believe that Lindley and Ortgies constructed the straw-man
argumentation on purpose, with deliberate intent to mis-represent my statements and my "implicit
premises". Rather, I believe it has been a serious and ongoing misunderstanding,
where they (probably) started formulating
assumptions and counter-arguments even before reading beyond the printed first section of my article.
(We must remember, there was a three-month hiatus between that and the printing of the second section: February
to May, 2005; and discussions and debates were already loud and heated in various Internet groups before
May.) Their first impressions against the thrust of my work were already locked down, ahead of consideration
of my entire article, including its web-only portions. They were already
casting this off as foolishness, taking my argument as if had little of substance in it beyond speculative
graphology. The evidence as I presented in part 2, and in the web-based "Supplementary Data", was not allowed
to come into the discussion with much (if any) weight.

When it came time to build their counter-argument into an article, they mis-stepped, and it compounded
the problem: they did not bother to ask me if their structure
(listing their guesses at my "implicit premises", and then concurring with or dismissing each one in turn)
made any sense. At least, they could have confirmed with me what I actually said in all of my article,
and what I would take to be my own premises, but they didn't.
Because they then got so many of those premises wrong, misunderstanding and then mis-representing
what I said, point by point, how could it be anything but a straw-man argument? They have been arguing
against things I did say, and things I do not believe, all along. Again, probably not deliberately or
maliciously! Because they didn't get what my paper actually said, or the evidence
I've presented (far beyond graphology), their counter-argument doesn't put up a fair or valid rebuttal.

There is more to come here soon, including my notes from playing
all the way through the WTC using Lindley's recommended "Bach-style" temperament from this article.
The musical/historical point is to find a suitable temperament scheme
to play this particular book, isn't it?
And since his suggested temperament is presented in the article's Appendix
with no given evidence (either within or outside the WTC) for the choices
within its intervals,
how does it stack up musically against mine that at least uses this WTC title-page drawing
(among other things) I've cited as evidence? Ignore the drawing as allegedly non-evidence, label
such pursuits as "daft", and then just make up something else...does it work?

The overall thrust of this article's argument

The presentation by Lindley and Ortgies boils down to the following points:

There is insufficient evidence that Ortgies and Lindley are willing to believe, about the way(s)
Bach personally tuned harpsichords.

Their method of examining and testing evidence (but only the evidence they choose to accept!)
has to be the only method that's considered valid...at least, to them. My additional methods of examining
and testing evidence (playing the music, measuring scale offsets against 1/6 comma standards,
analyzing harmonic and melodic motion, etc) aren't engaged by their article.

In lieu of proof/disproof, we're given a roster of patronizing scoffing (including the
words "outlandish" and "daft"), showing that they
will not deign to engage Lehman's "brilliantly written" position (their stated assessment).

In place of the temperament I recommend from Bach's drawing,
they present a made-up temperament of Lindley's on no presented evidence,
as their recommendation of "Bach-style" tuning. If according to them there's insufficient evidence
to make such a determination (i.e. that mine is somehow wrong or impossible for Bach), where
does theirs come from? If their point is to assert that Bach's tuning styles are unknowable,
with a skeptical agnosticism here, why should we replace my hypothetical temperament with theirs?

On careful reading, their article doesn't come across as a soundly reasoned or scholarly
rebuttal, proceeding by incontrovertible facts.
Rather, it comes across as their dismissive opinion (i.e. a personal position paper)
that my hypothesis is incredible, and as a scarcely disciplined rant that they personally
don't fancy the possibility it could be historically correct.

Repeating a few of my comments from the October section, above:
this article offers no reaction at all to my five
"web supplement" files that are integral
parts of the original Oxford article; as if only the on-paper sections of that
article are worthy of consideration?
I notice also that it hasn't engaged any of the technical material from my printed part 2,
which has been in print for 16 months (May 2005 issue of Early Music, and downloadable for
free here). The reactions are almost entirely to
the presentation in part 1 (February 2005),
which is only a small portion of my argument; and which evidently isn't fancied enough to
take the rest of the presented argument seriously.

They have not acknowledged or engaged any of my 2005 article
from Clavichord International, either, in its clarifications of the material;
have they read it? (It's more widely available on the web
now, to facilitate any such discussion!)
Nor have they offered here any musical or historiographical
reaction to any of my (or other people's)
recordings that use my proposed temperament;
isn't this too a dodging of potentially interesting evidence, on their part? It's
as if they made their minds up by looking
only at part 1 plus
my February 2005 comments on HPSCHD-L,
and then stopped listening to my further corroborations and explanations of my points.
Who's dodging unwanted evidence?

As of November 13 2006,
in the TUNING-L discussion
group,
Dr Ortgies is still stating his position as follows:
"Until today there is not a single piece of evidence, that allows us to
state which temperament Bach might have preferred at any time. This is
also the view hold within Bach-scholarship." Ortgies speaks for the whole body of "Bach-scholarship"?
What about my Bach-scholarship, or other people's who happen to agree that my work has merit?

See also his dismissal of my work as allegedly falling into "the fallacy of sentimens",
also November 13,
here.
He presumed in that posting to speak on behalf of correct scholarship:
"It is 'cold' reason, logic and application of scholarly methods which
tells us, that Lehman's claim is an unsubstantiated speculation -
whether we like his temperament, regard it as useful, or not." ...as if my work doesn't use
"scholarly methods", reason, logic, or a dispassionate-enough detachment from the material. Patronizing!

He is, of course, entitled to his opinion about that. It doesn't help, though, that his own
Early Music article then comes across as anything but dispassionate, and that it falls into variously
fallacies of reasoning and logic itself. He and Lindley in their article are the ones presuming to
judge such things, by the only viable standards....

The E-G# / Ab-C placement, and Neidhardt and Sorge

In the page of
errata posted by Dr Ortgies, he and Dr Lindley have amended two of their article's
points as follows:

(revision) "(c) Nor indeed did Neidhardt ever countenance tempering E-G#
more than Ab-C in any tuning that he recommended for use in any kind of social context whatever
(i.e. at a court, in a large city, in a small town, or in a village). [endnote 10]"

Oh, come on!

That is to say: Drs Lindley and Ortgies are willing to deal only with the small sets
of temperaments that Neidhardt singled out in 1724 and 1732 specifically for various "social
contexts" (their term!); and all the rest of Neidhardt's published temperaments within
these same two documents can be fluffed off as allegedly not being for "actual musical practice".

How? Why?

The E-G# and Ab-C point is not merely a matter of wording a few sentences more carefully,
to cover over their assertions where they were caught saying something that is crossed by the published
evidence (see my October section, above). It's obviously a play to try to save the skin of their article; but the
issue it's more crucial than that, for technical and musical reasons.
This placement of the note G#/Ab is fundamentally important to the
overall shapes of temperaments, the handling of sharp-based or flat-based music. This is of course
assuming a smoothly organized progression around the circle, from this point of placing the G#/Ab
at some reasonable position between E below and C above.

But, in the context of their argument here, Lindley and Ortgies are simply not willing to engage
the possibility of 18th century musicians using a wide E-G# "in actual musical practice"
"in any kind of social context whatever"! Their new over-the-top assertions here, while correcting
the factual problems of their first versions, make their tone seem even more dismissive in the
technical point their article refuses to deal with.

My point here is: Sorge (in 1758) and Neidhardt (in several of the 1732 temperaments) both
published schemes where this harmonic shape does occur, having E-G# wider than Ab-C.
These are documented 18th century formulas, unequivocal in the math.
It's Lindley and Ortgies who are tossing these out the window with an allegation that
they weren't serious temperaments for actual musical practice, in whatever "social contexts"! These
temperaments don't fit into their argument as evidence that they're willing to take
as a desirable musical feature; even
though this particular Sorge temperament (from 1758)
is given as their own Figure 4, right here in their article! Sorge recommended that
temperament for use especially in Chorton organs; the performance and improvisation of church
music presumably being a situation that is both "actual musical practice" and a "social context".

How do Ortgies and Lindley propose to "know" that Neidhardt and Sorge weren't proposing such schemes for
real musical use, or that we're supposed to take those as mere speculation, today? Simply
because they say so, and we move on as if they've proven it?
Their article doesn't establish any rules by which we're supposed to assume
that Neidhardt and/or Sorge were joking, or making up merely impractical speculations!

Their article
tells us only that Ortgies and Lindley themselves will not engage such temperaments that have E-G# wider
than Ab-C; and it's a convenient play to dump mine into the rubbish bin, simply by moving the
goalposts as they have done. Temperaments with a wide E-G# are outside their penchant to
take seriously...and this is propped up with
a weak/erroneous claim that Neidhardt and Sorge allegedly didn't take such things seriously either.

Nor have they provided any evidence of Bach moving such goalposts,
with an opinion one way or the other, or shown how a wider Ab-C is somehow better for his music.
Isn't that the point of fashioning credible temperaments for Bach's music, showing how and why
the choices of interval sizes are made?

By the way, their unchanged phrase "by as much as Dr Lehman says Bach always tempered it" is problematic
for another reason, being a straw-man misrepresentation of my article. My article is principally about
playing the Well-Tempered Clavier on harpsichords and clavichords, and no such "always" is
any central part of my argument. And yet, Lindley and Ortgies seek
to knock this down through appeals to organ-tuning treatises, in social contexts where Bach's preludes and fugues
were not the music to be played? Ortgies's own hypothesis is apparently that
nobody in German-speaking lands at that time played such repertoire in church anyway,
under pain of being denigrated as a "paper organist" (i.e. instead of improvising
fresh pieces); following that line of logic for the moment,
how do organ treatises tell us anything about the WTC, since allegedly that music was
not to be played there liturgically? (I must confess that I have personally played parts of the WTC on
organs in church services, and have recorded one of these preludes
and fugues on the organ to demonstrate how my proposed temperament handles it.
Too bad for me, in choosing to do another daft thing as a "paper organist".)

Let's summarize this batch of remarks about E-G#, etc.
By refusing to take Neidhardt's and Sorge's use of a wide E-G# as musically practical, Lindley and Ortgies are once
again simply refusing to engage my argument.
Rather, their own assertions about Ab-C
needing to be the biggest are (as in Lindley's New Grove articles) simply presented as the only
viable game in town, and the refinements are due only to the various sizes of the towns.
"Bach-style temperaments" must be pressed into their expectations. They are stuck
within that paradigm they've built for themselves!

They don't fancy it personally as a historical possibility, and this carefully spun appeal to
selective Neidhardt and Sorge material is one of their fallacious arguments
set up beside the material: looking like a refutation, but misleadingly so.
This is not to allege that they were deliberately being misleading; I believe they are simply
unwilling and/or unable to engage my argument from inside the sound of the temperament
style being discussed. The design with a wide E-G# is so anathema to them that they refuse to
consider this argument with such a "what-if" premise, even hypothetically. Neidhardt and Sorge are
then roped in, wrongly, to bolster that refusal.

There is even a triumphant bit in this article
that Lindley and Ortgies have been able to find "an alternative" reading of the drawing, suiting their
own conclusion (well, since it's a foregone conclusion it's really a premise of theirs!)
about needing to have a wide Ab-C:

"(p 617) (d) One can readily derive from the decorative loops a scheme of temperament in which Ab-C
is tempered more than E-G#. The appendix will show this." (p 619, appendix) "In fig. 5 this interpretation
is reduced to a musically feasible numerical scheme without involving any gradations finer than half of the
arbitrary unit. Here the pattern of relative sizes among the 3rds involving chromatic notes conforms,
better than in Dr Lehman's scheme, with the kind of evidence referred to under points (b) and (c) above."

Sure enough, the widest major 3rd in figure 5 is Lindley's usual one, and the one he places at lower
left in all his diagrams (influencing the way readers might expect it to be the farthest-out
from common harmony): namely, Db-F (enharmonically C#-E#), here sizing up at 9 1/2 units.
That is merely the same old Lindley argumentation over again,
as seen also in his New Grove articles et al,
and which I challenged directly in my February 2005 section, thus:

"(p 12, February 2005) Unfortunately, the expectation that Db and Ab will be pitched expressively low
is one that has been read into the music from Lindley's extensive experience with the basic
'workmaster' shape, and from his own background as a developer of new temperaments within that same
paradigm. And it does not necessarily have anything to do with the way Bach himself did organ
maintenance, or worked at stringed keyboards. This dictionary entry also has a glaring omission.
Where the temperament examples are provided, no temperaments are given that would cross
Lindley's observation about the quality of C#-E#. Most remarkably, Sorge's 1758 temperament with
the Bachian harmonic shape (see below) would be welcome in the New Grove discussion, at least
to show that 18th-century opinion was not unanimous about these shapes. Lindley had included it in
his brilliant 1987 article, noting especially Sorge's own assertions about this temperament's
suitability for Chorton organs with Cammerton orchestras. Lindley's New Grove
assertion about the necessarily worst C#-E# major 3rd therefore comes across as the only possible
game in town, and the only thing that really varies is the rusticity index of the town (following
Neidhardt's various formulations). In short, the familiar workmaster shape (in various proportions)
is presented as the only viable alternative to equal temperament. Temperaments that peak at E
major are not judged as presentable."

Lindley's argument (back to Early Music, November 2006, now)
is just a restatement of his own position, showing no willingness to change it.
Temperaments that peak at E major are still not given serious consideration,
as historically-feasible alternatives in musical practice. The only
"kind of evidence" he'll show us for serious practice
is that which suits his own generalization: about Ab-C (and Db-F)
needing to be wider than the major 3rds near E-G#.

Well: what's to stop
Johann Sebastian Bach from having done something that doesn't fit Lindley's and
Ortgies's expectations? (Or, for that matter, Neidhardt's or Sorge's expectations either: as in
being differently shaped harmonically, or having nothing to do with published
court/city/town/village distinctions?)
What's wrong with my presented argument,
either musically or in the manner a temperament style might be written down, in Bach's book
about tuning, except that it crosses Lindley's and Ortgies's expectations?
What's to stop Bach from doing something practical, simple, suave, and colorful
in his music--as my hypothesis suggests he did? Let's borrow a fine sentence of their own
from page 613 of their article: "The '48' are presumably the work of an expert and musicianly
tuner. We agree."

Other problems of fallacies, topic-dodging, and tone
in the Lindley/Ortgies article

First, it should be pointed out: this article by Lindley and
Ortgies doesn't offer evidence that they have tried out my proposed temperament in
extant compositions, or in improvisations, for any extended period of practical use.
(If they've tried it out seriously, they haven't said so.)
As their endnote #15 they offer only an anecdotal remark from a prominent
harpsichordist, who didn't fancy attempting it for one particular concert.
And, that remark in question was third-hand
or fourth-hand hearsay, not a direct quote: I recognize it from someone else's internet posting to
HPSCHD-L. The harpsichordist involved, Davitt Moroney, has remarked about that specific
anecdote in HPSCHD-L postings on
Nov 11 2006 and
Nov 12 2006...expressing the wish that Lindley and Ortgies had checked with him as a courtesy before
roping him into their endnote.

Also, they have presented no evidence that would
refute my hypothesis about Bach's actual practices on harpsichords or
clavichords. They merely assert that my
procedure is "daft" to use Bach's drawing as evidence ... and then go on to assert a
different "Bach-style" tuning of Lindley's own, for which no evidence is given. (That one in particular closely resembles
Thomas Young's #1, of 1799/1800, but with the tempering of the naturals relaxed
to be less than 1/6 comma. This gives
it room for tasteful adjustments around the sharp/flat side of the circle. Why?)

The structure of their article sets out what they call my
"implicit premises", in italics, to which they respond. This process
is substantially a straw-man argument
against my article, since I did not have those particular premises; they have only read
them into my work, on the way toward knocking down the ones they don't agree with.
I stated my explicit
premises, for the record, in May and June 2006 in the public forum
HPSCHD-L. These are available in my
digest at http://www-personal.umich.edu/~bpl/faq-hpschd.html
, or in that list's web archives. I
also posted them for easiest reference at the page
http://www-personal.umich.edu/~bpl/larips/history.html
. Ortgies and Lindley should have had plenty of time between May 2006 and October 2006 to clean
up their list of "implicit" premises next to those (i.e. to continue revising their article
pre-publication), had they been so inclined....

Here are my remarks about some of those straw-premises
(presented here [in brackets]), and about the commentaries on them.

This launches the series of straw-man points, misrepresenting my article on the way to presenting
a superior disagreement with it.
I did not state anything so strong or bald as that "always the same"; and indeed,
my own conclusions in that direction remain tentative (especially as Bach obviously did not
have absolute control over the organs he played early in his career; I never said he did).

Here is my most relevant paragraph, showing what I actually wrote. This is among a page of
my remarks about other specific pieces outside the Well-Tempered Clavier, and showing
how this temperament solves the long-standing problems in them:

"(page 212, May 2005) Judged from the substantial evidence of his music, did Bach learn, discover
or develop his specific tuning method by his early 20s, and continue using it for the rest of
his life? I believe it is likely, from the perspective of playability in all his music,
and from the inexhaustibly expressive resources of the temperament itself. I understand the
dangers of speculating that this temperament might reach far on either side of 1722.
But, on hearing how well it works in practice, I see no reason why Bach would ever have
discarded such an effective solution for his musical purposes. It allows his music to sound
beautiful, richly layered, and continuously engaging through balanced contrasts."

To be clear on this: Bach's earlier and later music works so well in the temperament
I have derived from his drawing, I am confident that Bach had some method to make
that music sound reasonable at the times when he wrote it. It didn't necessarily have to be
exactly the same as this; but I believe it was some method(s) allowing his music to be
played with pleasing musical effects. Why would any composer, Bach or otherwise, write music
that sounds bad on purpose? My contention is that he didn't.

In another important distinction here: my context makes it clear that this point was
a tentative conclusion, reached after months of studying, playing, listening, and writing.
That conclusion was (and remains) surprising to me also, but that's the direction in which I
believe the extant musical evidence points ("Judged from the substantial evidence of his music...").
That conclusion has changed the ways in which I approach
Bach's music as both musician and scholar. One example (among others)
is my suggested purpose that Bach supplied the four Duetti as tuning-test
pieces for organs; an idea that I believe is new to the Bach literature.
(Hence, my analogy to the "Rosetta Stone": for any willing to follow this musical adventure where it leads, taking this temperament seriously, it can unlock compositions that had seemed difficult to comprehend before. It can also change the way musicians and scholars think about the roles of keyboard temperament within music,
being sometimes an integral part of the music and not merely clothing to be applied willy-nilly later...if
they're willing to let actual sound be part of musical analyses, not merely treating intervallic
relationships from inside an abstract model of equal temperament!)

But, what have Lindley and Ortgies done here with this conclusion?
They have miscast (or misunderstood) it as a premise ("Bach's tunings of keyboard instruments
were always the same"), an overstated premise at that, and one
they can conveniently disagree with...since they don't engage much of my material from
the article's part 2 or the "supplementary files" anyway.

"(p 613) [He tuned certain 5ths pure, tuned others just like in equal temperament, and tempered all the
rest—including C-G-D-A-E—exactly twice as much.] ... There has never been any
musical reason to limit the choice of size for C-E to only two possibilities:
the equal-temperament size and the size yielded by tempering C-G-D-A-E twice as
much as in equal temperament. Musicians might well prefer an intermediate size."

As I have already presented in my article, the musical
reasons for those particular sizes are strong ones: principally the extant 18th
century pedagogical materials by Tosi, Sauveur, Telemann, Leopold Mozart and
others! (See my discussions of the
55-note division of the scale, which aligns with a general common scheme of 1/6
comma meantone.) I might add several
further bits of corroboration: the scaling in the construction of wind
instruments to play most easily in this 1/6 comma milieu (which I am exploring presently);
and the treatment of this same "equal-temperament size" (i.e. one schisma) by both Neidhardt and
Sorge as a theoretically indivisible unit, in practice.

"The focus of this book, however,
has not been on the various keyboard temperaments, but on what non-keyboard performers should
do or can do. String players, wind players, and singers all have some degree of flexibility
as to where they place the notes of the scale. Their great advantage, as noted again and again
throughout history, is that they are not obligated to put the notes--especially the accidentals--always
in the same place, but can choose a placement that fits the harmonic context of the moment.
Yet, because these musicians must sometimes perform with keyboard instruments, it makes sense
for their systems to be somewhat compatible."

Certainly, some musicians might prefer some other "intermediate size" intervals on their
naturals, instead of a 1/6-comma basis; but (I believe) the burden of proof is on Lindley and Ortgies
to demonstrate that Bach would have had sufficient reason to do so. If Bach had wanted
something substantially different from a
two-to-one ratio of tempering amounts, recognizing them by ear and by wrist action on the
tuning pins, mightn't he have found a way to draw his diagram differently? (That's assuming
that it is a diagram, of course, which Lindley and Ortgies won't even bring to the table,
but simply dismiss as a "daft" procedure.)

"(p 613) [He had a secret mathematical scheme which was passed down by word of mouth
or by making exact copies of a certain drawing.]"

My article was quite clear that I believe it's not a mathematical or theoretical
scheme by Bach, but rather a practical notation by him of something he used. I presented
historical corroboration (chiefly from CPE Bach's report) that
Bach did not go in for speculative mathematics!

But, what do Lindley and Ortgies do here?
They first set up a straw-point that I allegedly believe it's some merely "mathematical scheme",
and then they present reports from Mizler and Forkel that would contract this straw-Lehman.
I happen to agree with both the Mizler and Forkel quotations, that "Bach did not get involved
in deep theoretical treatments of music," and that "mathematizing would not ever have led to success
in ensuring the execution of an unobjectionable temperament". Those quotes both do well to support
my point that it was a practical/intuitive scheme away from any manner of calculation,
and that Bach wouldn't have approached it with any mathematical rigidity! I believe he tuned it
by careful musical listening and experience...and then wrote it down in a way that made sense to him,
that is, non-mathematically.

"(p 614) [There is only one musically feasible way to interpret this alleged evidence.]"

I said no such thing; my aim has been merely to present a plausible
reading of the drawing that gives Bach the greatest possible benefit of the
doubt, i.e. one that sounds colorful, smooth, interesting (admittedly all value judgments...),
and solves the outstanding practical
problems within his music. This is not principally
a mathematical scheme. I believe it is
a simple practical process of slightly flattening certain notes at the
harpsichord tuning pins, during everyday work. My mathematical presentation merely
measures the results, to explain the thing to an expected modern standard of thoroughness!

There are no calculations needed to set up this keyboard temperament in practice, but merely a process of
nudging the pin a little bit, or approximately twice as much, as indicated by
taste and experience. It is easy to do this in under fifteen minutes for the whole harpsichord,
by ear. I've been doing it several times a week, for more than two and a half years now. It's not
complicated. I have explained
all this more fully in an April 2006 article
"Bach's art of temperament" available at http://www-personal.umich.edu/~bpl/larips/art.html
.

By the way, does there exist a distinction between "alleged evidence" and "questionably credible
evidence", or perhaps "questionably useful evidence"? Bach's drawing is evidence of something,
existing on that title page of the WTC: evidence at least that he took the time to draw it. It's not
"alleged" evidence.

What are some of the most notable differences (to me) between this drawing and Bach's "decorative" curlicues
elsewhere?

It's drawn much bigger.

It's on the title page of a book about tuning, a book that presents a systematic approach to handling
all 24 major and minor keys.

It's jammed up against, and overlaps, words that say the subject of the book is keyboard tuning.

It has asymmetries among right, left, and center that look as if they may be deliberate, not
merely whimsical; the anomaly suggesting that there may be an important message here. On
this title page where everything else is centered across a mirroring midpoint, right to left, why is
the drawing at the top so asymmetrical in the types of loops it employs?

But, I have addressed such things in my FAQ pages, almost two years
ago already.

"(p 614) [The secret is
revealed when one looks at the loops upside down after having noticed the small
letter 'c' next to one of the loops.] The alleged small c is a kind of serif applied frequently in Bach's
handwriting to certain capital letters...."

I already addressed this point in endnote #63 of part 2, in
the original article, printed May 2005. I have also explained it more fully at my FAQ page, early in 2005 and
readily accessible to Ortgies and Lindley:
http://www-personal.umich.edu/~bpl/larips/faq3.html
. The primary reason to read the
naturals at that place in the diagram is not
because of this putative c, but because the first notes to be tuned
are (I believe) the diatonic notes of the C major scale—therefore starting the
series with F-C-G-etc. My reading of
the diagram would be exactly the same, even if this "small c" were altogether
deleted. Again, "Bach's art of temperament" explains this perhaps
better than my Early Music article did.

"(p 616) [Significant
documentary support for the claim that Bach always tempered E-G# by as much as
shown in fig. 3 is to be found in some remarks by Sorge ...] Neither Sorge nor Neidhardt ever
countenanced tempering E-G# (for any reason whatever) by as much as Dr Lehman
says Bach always tempered it. Neidhardt
would never countenance tempering E-G# more than Ab-C."

This is the weakest point in the Lindley/Ortgies article:
because it is factually mistaken! Neidhardt's 1732 publication includes 21 sample temperaments,
illustrating methods by which one may develop them. No fewer than five of
these 21 include an E-G# that is tempered as much as, or more than, mine!
This document is readily available at
http://harpsichords.pbwiki.com/Tuning
, both in facsimile and in a reliable transcription.

Neidhardt here begins with a Pythagorean scheme, placing the
entire Pythagorean comma in C-G; and the resulting E-G# is a full syntonic
comma sharp. He then presents twelve
example temperaments in which the 1/12 bits of comma are distributed variously,
emphasizing the qualities and varieties of 5ths, "Quinten-Circuls". His next chapter presents the concept of
creating temperaments by major 3rds, showing how to work at this experimentally,
and it begins with three sample
temperaments: all three of which have an E-G# as wide as (or wider than!)
mine. He continues with five more
practical examples, emphasizing 3rds.

I have already mentioned four of the five here; but the most
telling of all is back among the twelve "Quinten-Circuls". It is #11, the one that includes three 5ths of 1/12 comma wide, and has the following most prominent
feature: the E-G# interval is the same size as mine, the Ab-C is much smaller
(i.e. the same size as in equal temperament), and the C-E is smallest of
all. This temperament, even without its
four rougher brethren in the same document, sufficiently negates both the
Lindley/Ortgies assertions here! Its
complete pattern, for the record, is:

C 1 G 2 D 3 A 1 E -1 B 1 F# 2 C# -1 G# 3 D# 1 Bb -1 F 1 C.

The tempering within E-B-F#-C#-G# totals 1 (i.e. 10 out of
11 units sharp, where 11 would be a full syntonic comma). G#-D#-Bb-F-C totals 4 (7 out of 11 units
sharp); C-G-D-A-E totals 7 (4 out of 11 units sharp).

As noted above, in my section "The E-G# / Ab-C placement, and Neidhardt and Sorge", they
have attempted to correct those several erroneous statements to better readings.
Those altered sentences merely create different problems, however! And, if they had taken
those particular Neidhardt and Sorge temperaments as serious musical resources in the first place,
it seems to me they wouldn't have written such dismissive and misleading sentences that needed to be
corrected as errata.

"(p 617) [Bach's occult
mathematical invention is of such beauty that it suits uniquely well the extant
music of Froberger (1616-67) and Frescobaldi (1583-1643).] This finding is historically outlandish (and
therefore undermines the hope which one might have entertained about the value
of Dr Lehman's aesthetic judgment as ancillary evidence for the taste, in this
regard, of musicians in bygone centuries)."

This is such a gross overstatement of my position as to be
absurd. It's one of the most patronizing barbs and outrageous pieces of straw-man argumentation (among
several others) in their article. I did not propose any such thing. Therefore,
their gratuitous digs against my alleged taste are both misleading and inconsequential.

Rather, my remarks about 17th century music known
to Bach (including Froberger's and Frescobaldi's) were in passing, as I described
my process of personally testing my temperament thoroughly...playing through such repertoire to hear what it sounds
like (i.e. entertaining the notion that Bach might have done the same). Nor did I use the word "occult" anywhere in
my article, or call this temperament a "mathematical invention"; I consider
this temperament a simple picture of adjusting notes in practice at a keyboard,
not a "mathematical" construction.

Furthermore, there is no reasonable excuse for such
presumptive comments against my "aesthetic judgment" or against my own
practices as harpsichordist; in e-mail discussion with Dr Ortgies and others in
spring 2006 (23 May 2006 on HPSCHD-L),
I made it quite clear that I maintain one of my several
harpsichords in or near meantone temperaments all the time, to continue to play
16th and 17th century repertoire.
Furthermore, one of my own doctoral recitals in harpsichord (1994) was entirely in 1/4 comma meantone,
and I still play those compositions in or near meantone because of its appropriateness.

Perhaps they are over-reacting to several of my remarks in
the booklet notes of my organ CD set? I
wrote the following:

"This system turns out to be an
excellent tuning solution to play all music, both before and after
Bach's. It is moderate enough for complete enharmonic freedom, but also unequal
enough to sound directional and exciting in the tensions and resolutions of
tonal music. I have chosen the music of this CD set to demonstrate the
expressive resources and contrasts available within this system...and of course
for the sheer beauty of the resulting sounds on this new organ! (...) Among his
many adventures, Johann Jakob Froberger had been a court organist in Vienna
from 1637. I chose to record this particular Froberger toccata from 1649 as it
is especially wayward harmonically. Liturgically it was probably played with a
somewhat quieter registration, in its designated function for communion: but it
can also be a wild and exciting piece when played robustly. How were the
keyboards tuned, to be able to visit the triads of both B major and A-flat
major within a G-Dorian context as here? This composition uses 15 different
notes in its course: Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, and D#.
If Froberger's basic tuning was anywhere near classic meantone (based on pure
major 3rds), he either had some split keys on the instrument, or he and his
listeners put up with occasionally raucous moments from the enharmonic
misspellings."

But again, there's nothing here in my remarks to resemble
their overstatement, "suits uniquely well the extant music of Froberger
and Frescobaldi"; I was simply pointing out that I've recorded a Froberger
piece which arguably sounds good on this organ, and which presents problems when
played in other twelve-note temperaments!

"(p 617) Dr Lindley...has
proposed to Dr Lehman 'a joint presentation ... for the purpose of ...
demonstrating aurally to the same people on the same occasion' their respective
musical results."

Well, there are certainly some important things here that Dr Lindley isn't telling us.
Since he's brought it up as a topic: that e-mail message from Dr Lindley to me
was on 16 December 2005. In it he suggested that such a play-off between the two of us
should demonstrate the following three compositions from WTC book 1:
the E major, A-flat major, and C major preludes. He continued:
"Each of us will have the challenge, as a performer, of making a
virtue of the relatively more nervous parts of our respective tunings."

I replied to him immediately (19 December) as follows, agreeing that it
would be interesting: "For such an occasion, if it ever takes place,
I would have to insist additionally on the book 1 preludes in C# major,
F minor, and Bb minor. It would seem only fair that each person
gets to select the same number of preludes.
These three have such prominent spots on downbeats, and often
with wide spacing to major 10ths, of the classic problematic
intervals. ... One somewhat unexpected thing I have found about
my E major, in the Bach repertoire and not only in WTC, is that
its sound is especially lyrical in the way it emphasizes
smooth melodic motion. Quite the opposite of any 'nervous' effect."

At this point, where I had insisted that it should be a fair contest with each
person picking half the material for the test, Dr Lindley did not respond any further.

He also had the advantage, at that time, of knowing my temperament's layout but without
telling me his. Now, having played through the WTC in Dr Lindley's temperament (during October-November
2006) to hear what I'd be up against,
I might do even better to select pieces that work even less well in his than
the C# major, F minor, and Bb minor preludes! Those are fine examples, to be sure, and would probably
be sufficient for such a play-off; but there are some
additional problems in (for example) the D# minor fugue, F minor fugue, F# major prelude,
F# minor fugue, G# minor prelude, Bb minor fugue, and more. I also don't like the
things his temperament does to the simplest keys of all: C, F, and G majors! They're made
to be so bland and unresonant....

My own recordings on
harpsichord and
organ
are intended for exactly this sort of comparative listening, putting them next to anyone
else's recordings of the same repertoire, and using any temperaments of their own choice.
Enterprising listeners can surely come up with some fair "taste tests" in this way,
to hear the effects and characters due to temperament.

An even better comparison for close listening, of course, is to have two harpsichords side by side, tune both
of them by ear in competing temperaments, and then play the repertoire directly on both, oneself.
This is in fact what I do all the time, with two harpsichords set up in my living room permanently.

"(p 618) [A significant degree of aesthetic and historical consensus is indicated
when the wife of a man who is passionately committed to a certain aesthetic
point agrees with him.] Caveat lector."

That one is so patronizing to the point of hilarity, I needn't comment further on it.

"(p 618) Dr Lehman's one-size-fits-all approach to 18th-century-style tuning is not only historically mistaken but also likely, we think, to distract one from noticing some relevant facts about church organs in those days."

The material presented in their section here is of course interesting and valuable, but it overlooks a main point: that my article to which they are reacting was primarily about harpsichord practice, and only marginally about organs (principally my remarks about Chorton transpositions).

"(p 620) The whole idea {of deriving temperaments from Bach's WTC title page drawing} is daft, but if he really wishes to pursue it, let him account for all the details ...."

The passage here is both patronizing and dismissive, while not providing any proof that Bach could not have done anything like this! This "daft" assertion simply indicates that my results and methods do not make credible sense to Lindley and Ortgies.

In this same section they present five other ways in which this same "daft" pursuit may be done, allegedly (in their opinions) as well as or better than my proposal; out-dafting me. Their point here, a reasonable one, is that my scheme is not a unique one; but they have already said that earlier in the article, and I never suggested that it was either a unique or unchallengeable reading. My "accounting for all the details" is in my long sections of musical and theoretical analysis. My article was never a mere look at calligraphy; more substantially, it has been rooted principally in musical practices and in methods of 18th century pedagogy.

Lindley and Ortgies here dismiss my work as nothing more than unbelievable "tea-leaves" reading (their term!); and they selectively overlook the actual evidence I have presented, which is Bach's music in analysis and performance.

"(p 620) The whole idea is daft, but if he really wishes to pursue it, let him account for all the
details and take into consideration all the 'hidden' German letters here and there in the upper part of
the title-page that can be detected upon consulting Tafel IV etc. in Grun's Leseschlüssel (if he
is not in a position to consult primary sources)."

Well, thanks, but that is (once again) overbearingly patronizing in the way it's worded. Let's turn
this same patronizing table around and view it from the opposite direction. If Drs Lindley and Ortgies
really wish to pursue the sound of my proposed temperament, to hear what it really does in musical
practice that they may offer musical reactions to it (even though Ortgies maintains that such
a tasteful procedure is non-evidence!), good resources are available. Let them purchase
my harpsichord recording of WTC excerpts et al, and
Peter Watchorn's complete WTC Book 1, to hear it (if they are not
in a position to do primary research on this by playing the whole WTC themselves).

"(p 620) Dr Lindley suggests the following as an expert and musicianly approach to setting a Bach-style unequal temperament on harpsichords (and pianos) at modern pitch ..."

This is all well and good, in practice, presenting a temperament whose overall shape closely resembles Thomas Young's scheme #1 (published 1799/1800), but with a lighter C-G-D-A-E, working out to approximately 1/7 comma each. This gives more wiggle-room around the sharp/flat size to create the types of E-G# vs Ab-C colorations that Dr Lindley asserts are essential.

The trouble is that he has not presented any evidence that such a scheme should be "Bach-style", but merely that it sounds "musicianly" and works for him.

If he would simply set up, for example, the Neidhardt 1732 "Fifth-Circle #11" as described above, and play through
the WTC (as I have done...), listening very carefully...he might come to the same subjective conclusion I have,
which is that this particular Neidhardt temperament sounds more "musicianly" and less problematic
in the WTC than Lindley's does!
Such a test is of course open to anyone who has two harpsichords, or a double-manual harpsichord, plus
the tuning and playing skills to go through the WTC: set up Lindley's and this Neidhardt "Fifth-Circle #11"
(a clearly-documented and authentic 18th century temperament) for direct comparison side by side,
and play the music. What should we do with his assertion that such Neidhardt temperaments were not
for practical use?

And (a small point), why do Lindley's tuning instructions start from the note E? Where do we (or Bach!)
obtain an E tuning fork or similar reference to begin, if we regularly tune our instruments without
any electronic aids? Granted, one might start with a standard C fork, build the prescribed E first,
and proceed from there...but the printed instructions do not suggest that.

"(p 621) (endnote #2) The late Dr Herbert Kellner believed that the design of Bach's seal,
with its seven points and five dashes (see illus. 3), proves that Bach tuned seven
5ths pure and tempered the other five, viz. C-G-D-A-E and B-F#. Dr Lindley has remarked
that if the design of the seal referred to music, then a more obvious meaning would be that
normal keyboards provide five chromatic and seven diatonic notes (....) These remarks by
Lindley led Stanley Sadie to reject an article which Dr Kellner submitted in 1985 to
The musical times."

This endnote also wastes space with an illustration showing Kellner's mapping of notes to the seal. So
what? I don't stand for Kellner's strings of forced coincidences either, and my article clearly
says so; nor is my article any defense of Kellner's outcome (quite the opposite, in fact!) or his methods.
Frankly, this endnote by Lindley/Ortgies looks to me to have two main purposes, both being poor sportsmanship:

To harness me in with Kellner's procedures somehow by oblique implication, as if this somehow will
increase disbelief in or scorn against my quite different work.

To express Lindley's apparent displeasure at not being invited to veto my article, pre-publication,
the way he was able to squelch a Kellner article so triumphantly.

But, this endnote #2 shows the same strategy as they use in this article's Appendix. They first assert that something is absurd, and then they devote space to showing how the absurdity should have been interpreted better if it
were viable evidence. (If Lehman were smarter, he would have done it this way and presented these
other allegedly more favorable results; but then again, if Lehman were
really smart, he wouldn't have done it at all but left the article in the rubbish heap, unsubmitted!
Er, thanks a lot.)

Come to think of it, why is the Appendix of their article the really substantial technical part of the thing?
Why didn't they organize their article so that this discussion of practical temperament (here in their
"Appendix") was the main section, and put their roster of straw-Lehman premises into an Appendix--if it
even needed to be presented at all?

Playing through the WTC in Lindley's temperament

My following remarks are from a practical test (October 2006): playing straight through the WTC book 1 in
Lindley's proposed temperament, and stopping to jot comments on paper whenever the music sounds harsh or
startling. This was done on a Flemish-style harpsichord, tuned carefully according to Lindley's
temperament given in the Appendix of this article. Wanting to react only as a listening musician,
I deliberately did these playing sessions before venturing into any numerical analysis (see further below).

Let it be said, in appreciation, that this temperament already sounds considerably better in this
music than Werckmeister 3, Vallotti, Kirnberger 2 and 3, Kellner, Barnes, and some others. On the other
hand, that's not saying much! Lindley's temperament merely tames down the same basic problem they
have, collectively, which is that the major 3rds B-D#, F#-A#, C#-E#, and Ab-C are too bright
relative to the other major 3rds. It also opens up F-A, C-E, and G-B to be less stable than they could
be--and this contributes to an overall blandness, plus an inability to settle into richer resonance. Its
moderateness is both a virtue and a downfall.

Play-through notes, piece by piece:

C major: seemed OK.

C minor: seemed OK, other than a few occurrences of Ab that seemed too low melodically.

C# major: as early as the first note of the prelude, C#-E# is harsh in this wide-open spacing.
The rest of this prelude and fugue exhibit other overall harshness, as well, on too many occasions to write
down individually. The whole thing was five minutes of cringing! Even the unaccompanied statements of
the fugue subject sound uncomplimentary, with their too-wide leaps of 6ths melodicaly.

C# minor: again, too many cringe spots to write down individually. The dominant triad is ugly, the
B# and Fx sound too high in their contexts, and the occasional E# and A# draw too much attention to themselves
as well.

F minor: prelude's middle of bar 6, downbeat of 9, middle of 16. Fugue's middle of 7, end of 24, downbeat 38,
end of 38, downbeat of 40, middle of 46, middle of 54. And, the final chord of the fugue beats more rapidly
than I'd like to hear, with its quick F-A wobble in this tenor-range spacing above the low bass F (sustained
major 10th).

F# major: prelude and fugue both too painful all the way through to enumerate spots!

F# minor: prelude's downbeat of 11, downbeat 18, middle of 20, end of 20, last chord. Fugue's downbeat 10,
end of 16, middle of 19, end of 26, end of 27, downbeat 31, last chord.

G major: OK, although the last chords of both the prelude and fugue seem too active. It's the end of the
piece, and seems it should settle better!

G minor: decent enough, except for the second beat of prelude's bar 10 and the fugue's middle of 22.

Ab major: prelude's downbeats of 24, 25, 28, 32, 39, 43! Fugue's many spots where Ab-C are played as an open
major 10th.

G# minor: the B-D# 10ths are rough as early as bar 1! Also the C#-E# in bar 12, F#-A# in bar 5, and the last
chord of the prelude. Fugue decent enough, except for the B# and Fx and Cx sounding slightly strange, melodically.

A minor: prelude's downbeats 11 and 16 don't arrive firmly as the C-E(!) is so sharp and unstable-sounding.
Fugue gives a general feeling of long, directionless blandness; there are no big arrivals to the tonicizations
of C, F, or G majors.

Bb major: excellent.

Bb minor: prelude's second beat of 3, second quaver of bar 8 draws too much attention to the weak part of the
beat. Other problems in second beat of 10, second beat of 12, middle of 14, downbeat 15, middle of 15.
Fugue's middle of 5, middle of 21, middle of 24 going into 25, downbeat 37, middle of 38, middle of 41,
downbeat 45, downbeat 47, downbeat 64, downbeat 69.

In the major-3rd bounds within Lindley's instructions (the beat rates of his major 3rds at A=440 pitch),
it is apparent that his scheme on the naturals
captures the range between 1/7 Pythagorean comma and 1/7 syntonic comma, inclusively.

Moreover, as I have noticed both in this play-through and in some using various Neidhardt temperaments,
there is a liability that is scarcely apparent on paper. This is a phenomenon when playing on harpsichords;
organs and pianos behave somewhat differently.
As noticed in the natural major 3rds here (F-A,
C-E, and G-B: these are the major 3rds of sizes 4 to 5. They hit the "no-man's-land" region between reasonable purity/stability (size 3 or less), and the rapid
buzz typical of equal temperament (size 7 or more). That is, they beat rapidly enough to grab extra attention (in musical contexts), but not vigorously enough to turn into an energetic blur--which happens at approximately
size 6 and above.

Again, this is no number-juggling, but a reaction from listening within ordinary
musical contexts, and specifically on harpsichords.
Measurements of major 3rds, from size 0 up through 11 and more, might look like simply
a continuum on paper; but, there is a region in there (around sizes 4 to 5) where the major 3rds in practice
sound especially harsh.

I understand that this might look like a paradox, but I encourage any skeptics to listen to this for themselves: major 3rds on harpsichords sound better when they are
either smaller than or larger than this rough effect in the middle range!
And, this happens to be the range of Lindley's naturals in the three simplest
major and minor keys.... [Here's a conjecture, aside: is this phenomenon in part behind the fact that
1/7 and 1/8 comma temperaments make such little appearance in the historical record? Because their
resulting major 3rds hit this raw and ugly midrange size, and therefore weren't much worth bothering
with in practice?....]

Here are examples of Lindley's layout as it works out in practice,
interpreted both as 1/7 Pythagorean comma and 1/7 syntonic comma. These are both based on the
beat-rate ranges given in his article, at A=440, and worked back to be displayed here
as fractions of commas:

"Bach-style" and "Handel-style" by Lindley

Compare the above with the following interesting temperament developed by Lindley, c1995, for the
Handel organ at St Lawrence Whitchurch
(London suburbs). It also resembles the batch of experimental "Bach-style" temperaments on the last several
pages of Lindley's Michaelstein conference article (1994-7).

And all of these are seen to be quintessentially Lindley-style,
within the algebraic methodology and premises of that Michaelstein article:

C-E must be the smallest major 3rd

Db-F must be the largest major 3rd

C-G-D-A-E 5ths are all the same as one another, but tempered less than 1/6 Pythagorean comma

E-B and F-C 5ths are the same as one another, but tempered less than the C-G-D-A-E 5ths (i.e. more nearly pure
but not yet pure); like the strategy displayed in Thomas Young's #1, 1799/1800

Hair-splitting microfractions of commas, at the various other 5ths; as if the basically indivisible
unit of 1/12th PC used in Werckmeister's, Bendeler's, Neidhardt's, Sorge's, and Marpurg's temperaments
is simply not fine enough for Lindley.

The publication of this "tuning specially devised by Mark Lindley for Handel's music"
is in Paul Nicholson's set of Handel organ concertos, recorded May-June 1996, issued
as a Hyperion CD set. Page 12 of the booklet gives the temperament. Remarkably, Lindley's split here uses
1/96th fragments of the Pythagorean comma!

C-G-D-A-E are 15/96th PC narrow, each

E-B and F-C are 11/96th PC narrow, each

B-F#-C# are 7/96th PC narrow, each

C#-G# is 3/96th PC narrow

Ab-Eb-Bb-F are 1/96th PC wide, each (imperceptibly of course, as noted also in a footnote of the Michaelstein article...)

It works out with the major 3rd sizes all hitting integers or halves, and the whole thing looks like this:

I should add: this temperament sounds terrific in that CD set, played on organ in Handel's organ
concertos. But also: Handel's music is not as harmonically adventurous as Bach's...so why should there
be any expectation that modern temperaments for these composers' music should be of similar shape
as one another?

The Hyperion booklet doesn't present any of Lindley's reasoning here, either (as the present article
about Bach also doesn't). Apparently we're just supposed to
take his idealized temperament style as solved, for Handel's music, on faith in Lindley's authority...and
if we happen to know about it, on Lindley's algebraic "lucubrations" (his word) at the Michaelstein as well.

It still all spins back round to the same set of premises in Lindley's personal preference,
by which Lindley is now judging my work to be faulty in that it disagrees with his! Historically
based preferences, to some extent, yes; but it has the same leaps of faith that he gives us in his
New Grove articles as well. Whatever fits his model outcome, as exemplified in these, is good; and
whatever doesn't fit is wrong.

Also of interest, at least to me, is the relative sizes of Lindley's naturals here for Handel.
They are basically 15/96th PC until we get to the F and E
endpoints, each spread out a little wider; that is, they are tempered at a size
between 1/6 Pythagorean comma and 1/6 syntonic comma (i.e. about 11/72 PC).
This same tiny region in between 12/72 PC and 11/72 PC is where the regular 55-division of the octave also lives.
So essentially, Lindley has put his "Handel" naturals onto the 55-division spots described by theories of Tosi, Sauveur, Telemann, Leopold Mozart, et al. I agree!
(And so do Thomas Young's two temperaments of 1799/1800.)
All hair-splitting aside, anything in this little region bounded by 12/72 PC and 11/72 PC
is historically plausible, to me, for the regular tempering of the naturals.
It's Lindley's treatment of the five sharps/flats that I don't agree with.
They're probably OK for Handel, but they're not high enough for Bach's music.

Ortgies's further commentary on his web site

As of November 5th 2007 (and probably considerably earlier),
Dr Ortgies's errata
and corrigienda page for his article has the following two paragraphs introducing the piece:

In it we discuss some of the historically wild and methodologically wrong speculations published last year (2005) by Bradley Lehman in his article in Early Music (Note 1), where he claimed to have discovered "Bach's temperament."

The Lehman temperament is of modern design. It is, like the one designed by the late Herbert Anton Kellner on July 7th, 1977 (i.e. 7/7/77 - so Kellner said), based on an imaginative interpretation of a small image from the year 1722: for Kellner, Bach's seal; for Lehman, the ornamental scroll at the top of the title-page of Part 1 of Das Wohltemperirte Clavier ("The Well-tempered Clavier").
Lehman is following previous musings, especially by Andreas Sparschuh, who published the ornamental-scroll idea on September 9th, 1999 (9/9/99) as a kind of practical joke to make fun of Kellner.

That's sort of amusing, as their article doesn't say anything about Sparschuh playing a practical joke at Kellner's
expense...or about any of the 7/7/77 and 9/9/99 number games. If this were important, why didn't Ortgies and Lindley
put it into their article? I didn't "know" about it until today, reading it there on Ortgies's page. 1977? Really?
I'd always followed Kellner's words from one of his articles, or maybe it was Kellner's web site,
claiming that he figured it out during the week of Christmas, 1975. But we digress.

Nor does their article explain what is meant here by "historically wild" and "methodologically wrong"
in assertions against my work. Wow. Window dressing.

And, Ortgies's page has links to "Full text" and "Full text (PDF)" of their article...but those links work only for
paid subscribers to the journal. Since they're not giving away what they actually wrote, the
errata
and corrigienda and the
abstract
will have to serve some of their readers who will see only those bits instead of their article.
Hmm. My work is all downloadable for free
as a courtesy services that Oxford provides to all their
authors: a toll-free link for use on an author's own web site, for promotion of the work. Why haven't Ortgies
and Lindley availed themselves of their free links, to bring their arguments before the general public?

November 2008: Lindley has a short bit of correspondence published in the current issue of Early Music. It
includes this sentence:
"I think it was kind of Ortgies to omit from his review any mention of Duffin's endorsement of the notorious idea that Bach had a secret mathematical formula for tempered tuning which he conveyed cryptically in the decorative loops of the title-page of Part 1 of the '48'."

Yes, that would indeed be preposterous and far-fetched, especially because Bach did not go for any "dry, mathematical stuff". A secret mathematical formula, encrypted? It shows that Lindley is (after more than three years) still stuck in his own straw-man argument where he apparently doesn't understand a main point of my article! I never said it was any "mathematical formula", secret or otherwise; nor did I say Bach was being cryptic with it.

Bach's drawing, in my opinion, is a straightforward and practical diagram to get the work done without any calculations. Nor do I require Bach to have understood anything in theory about commas, or mathematical constructions. To me, the loops in Bach's diagram indicate a simple narrowing by "double" or "single" amounts from a pure 5th, by experience and listening, not by any secret or overt mathematical process! The thing takes less than three minutes to do, accurately enough in practice, sitting at a harpsichord with a tuning lever. How difficult is that point to understand?

Perhaps Dr Lindley (if he ever sees this web page that rebuts his preposterous published comments)
should take a look at my
lecture notes from 10/22/08.
In that presentation I explained the theory without requiring the audience to do anything mathematically difficult,
or even to read music. I showed how the principles arise from the necessity of playing more than two dozen
differently-named notes within that book of music, Das wohltempirirte Clavier. I demonstrated the easy
hands-on process of tuning, without requiring Bach or anyone else to calculate commas. The presentation didn't
go into "the little C capitalization stroke" that was such a red herring to both Lindley and Ortgies, among
others; Bach's tuning sequence (according to my analysis) requires the same placement of C whether or not that little
stroke is taken as part of it! Are there any other silly little points of contention that are still tripping up Lindley
and Ortgies from understanding the work they're trying to criticize?

August 2009:
In Mark Lindley's review of Patrizio Barbieri's book Enharmonic instruments and music 1470-1900
('A great microtonal survey', Early Music, xxxvii/3 (2009), pp.481–3), there is this sentence:
"The section on 12-note equal temperament includes a page and a half on this other topic (with an appropriately dismissive footnote about Bradley Lehman's hypothesis regarding Bach's use of an unequal circulating temperament)".

November 2009:Early Music, Bradley Lehman:
"Unequal temperaments circulate again" letter to the editor, Early Music.
A call for reasonable and valid argumentation in the field of temperament research - especially from Lindley,
whose published remarks in that journal have now mis-represented my work three times.

I still do not believe he is doing that deliberately; rather, it seems to me that he misunderstands
the thrust and the evidence of my whole argument, and he is still arguing against (and belittling) that
underestimation of it.

The letter is printed and published online in Early Music (Oxford University Press), February 2010, Vol 38 #1, p 170-171.
[PDF]
[HTML] It had been accepted for the November issue, but the entire Correspondence section was then delayed until February.